Research Problems in Harmonic Analysis and Partial Differential Equations

研究者计划研究Tomas-Stein不等式在球面上的极值问题，建立一个非椭圆相位振荡积分的Eschenhartz估计，并研究二维水波方程的Cauchy问题。前两个项目旨在了解振荡积分的某些方面，这些方面在欧氏空间中具有曲率的超空间的傅立叶变换的限制/扩展理论中很重要。Tomas-Stein不等式的极值问题是问是否存在一个函数优化不等式，使其成为一个等式。它还包括问题的特点极端化，如建立光滑性。第一个项目是关于一维和高维球面的Tomas-Stein不等式的极值问题。第二个项目的重点是研究一个振荡积分与非椭圆相位，这是在形式的Eschenhartz估计。它是由这些估计在薛定谔方程中的应用所激发的。第三个项目是研究具有表面张力的二维水波方程的Cauchy问题。众所周知，在以前的工作中，在接近这个问题的一个重要步骤是减少原来的方程组的一个合适的和等价的拟线性系统。因此，这一约化如何在发展适定性理论中发挥作用仍然是一个有趣的问题，所提出的研究将从多个角度产生更广泛的影响。首先，这些问题的调查在于接口的几个分支的数学，如分析和偏微分方程。因此，他们的解决方案将促进这些领域之间的互动。此外，这些问题涉及到数学技术的发展和人类对数学基本概念的理解。例如，托马斯-斯坦不等式是数学中一个基本运算--傅里叶变换的重要度量，它可以追溯到一个基本问题：一个无穷级数（在这种情况下是给定函数的傅里叶级数）何时是可求和的？ 这些问题影响应用科学和工程的方式的例子是多种多样的。例如，水波方程被用作描述表面波在河流或海洋上的传播的模型。对该模型的严格研究将为建模和计算模拟提供理论基础，这反过来又使研究人员能够深入了解一些破坏性的物理现象，如流氓波和海啸。最后，透过研究者的教学活动分享研究经验，将增加数学研究的认知与欣赏，并增进科学社群与社会的多样性。

The investigator plans to investigate the extremisers problem for the Tomas-Stein inequality for the sphere, to establish a Strichartz estimate for an oscillatory integral with a non-elliptic phase, and to investigate the Cauchy problem for the two dimensional water wave equations with surface tension. The first two projects aim to understand some aspects of oscillatory integrals, which are important in the restriction/extension theory of Fourier transforms to hyper-spaces with curvature in the Euclidean spaces. The extremisers problem for the Tomas-Stein inequality asks whether there exists a function which optimizes the inequality so that it becomes an equality. It also includes questions of characterizing extremisers such as establishing the smoothness property. The first project concerns the extremisers problem for the Tomas-Stein inequality for the one and higher dimensional spheres. The second project focuses on studying an oscillatory integral with a non-elliptic phase, which is in form of Strichartz estimates. It is motivated by the applications of these estimates in the Schrodinger equations. The third project is to investigate the Cauchy problem for the two dimensional water wave equations with surface tension. It is well known that an important step in approaching this problem in previous works is a reduction of the original system of equations to a suitable and equivalent quasilinear system. So it remains an interesting question how such a reduction will play a role in developing its wellposedness theory.The proposed research will result in broader impact from several points of view. First, these problems under investigation lie at the interface of several branches of mathematics, e.g. analysis and partial differential equations. Thus their solutions will facilitate interactions among these fields. Moreover, these problems concern both the development of mathematical techniques and the human understanding of the fundamental concepts in mathematics. For instance, the Tomas-Stein inequality is an important measure of properties of a basic operation in mathematics, the Fourier transform, and goes back to a fundamental question: when is an infinite series (in this case the Fourier series of a given function) summable? Examples of ways in which these problems impact applied science and engineering are many and varied. For instance, the water wave equation is used as a model to describe the propagation of surface waves on a river or the ocean. A rigorous study of this model will provide the theoretical ground for modeling and computational simulations, which in turn allow researchers to gain some insight into some destructive physical phenomena such as the rogue waves and tsunamis. Finally, sharing the research experience through the investigator's teaching activities will increase the awareness and appreciation of mathematics research, and improve diversity of the scientific community and the society.